MathOverflow Asked by Xindaris on December 6, 2020

Suppose we have an inverse system of compact Hausdorff spaces $lbrace X_i , varphi_{ij} rbrace_{iin I}$ and that each space has a presheaf $Gamma_i$ assigned to it in such a way that $Gamma_i(varphi_{ij}(U))=Gamma_j(U)$ whenever $ileq j$. Then $X:=varprojlim X_i$ has a presheaf $Gamma$ defined on it by $Gamma(U):=Gamma_i(varphi_i(U))$ where $varphi_i:Xto X_i$ is the map from $X$ as an inverse limit; this is well-defined since $varphi_{ij}(varphi_j(U))=varphi_i(U)$ whenever $ileq j$, which makes $Gamma_i(varphi_i(U))=Gamma_i(varphi_{ij}(varphi_j(U)))=Gamma_j(varphi_j(U))$.

In this situation, does Čech cohomology satisfy a continuity property? That is, is it true that $breve{H}^*(X,Gamma)=varinjlim breve{H}^*(X_i,Gamma_i)$? I’ve seen it claimed in some places, such as this question or even wikipedia’s talk page for Čech cohomology, that Čech cohomology satisfies some kind of continuity property for "nice enough" spaces, but I can’t seem to find any clear reference for this fact. The paper that question refers to seems to be concerned with a more general situation involving triangulable pairs, and I can’t fully make sense of it.

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- haakon.io on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Joshua Engel on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP